Simulation and the bootstrap

packages

library(tidyverse)

Simulation

• Sometimes you know the exact mathematical answer to a problem, eg:
  • \(X_1, \ldots X_n \sim N(\mu, \sigma^2)\), what is distribution of \(\bar{X}\)? (Ans: \(N(\mu, \sigma^2/n)\).)
• More often, though, you don’t:
  • if \(X \sim Bin(2, 0.5), Y \sim Bin(3, 0.2)\), what is dist of \(Z = X+Y\)?
• Simulation: generate random \(X\) and \(Y\), calculate sum, repeat many times. Gives you (approx) dist of \(X+Y\) without any mathematics!

Random numbers in R

• R knows about a lot of distributions, eg: norm binom pois exp gamma t chisq (type ?distributions in Console to see more)
• to generate random numbers from a distribution, put r on front of these; inputs are number of random values to generate, and parameters of distribution to simulate from.
• Examples:

rnorm(5, 100, 15)

[1] 111.96072  89.80331 113.74231  90.74167  89.86461

and

rpois(10, 3.5)

 [1] 5 2 5 5 3 5 1 3 4 0

Our problem: simulating once

• if \(X \sim Bin(2, 0.5), Y \sim Bin(3, 0.2)\), what is dist of \(Z = X+Y\)?

x <- rbinom(1, 2, 0.5)
y <- rbinom(1, 3, 0.2)
x

[1] 1

y

[1] 1

x + y

[1] 2

To simulate many times: - set up dataframe with space for each simulated value - work rowwise - do one simulation per row

Simulating many times

tibble(sim = 1:10000) %>% 
  rowwise() %>% 
  mutate(x = rbinom(1, 2, 0.5),
         y = rbinom(1, 3, 0.2),
         z = x + y) -> d
d

Distribution of sum

Make a bar chart rather than a histogram because distribution of \(Z\) is discrete:

ggplot(d, aes(x = z)) + geom_bar()

(Simulated) probability that the sum is at least 4:

d %>% count(z >= 4)

Only this much:

285/10000

[1] 0.0285

A sum of 5 is possible though very unlikely.

The bootstrap

Source: Hesterberg et al

• Sampling distribution of a statistic is distribution of that statistic over “all possible samples” from population of interest.
• “Plug-in principle”: sample mean estimates population mean, sample variance estimates population variance, etc.
• Also, sample is estimate of population (precisely, proportion of sample values \(\le x\) estimates probability of drawing value \(\le x\) from population, for any \(x\)).
• As long as your sample is representative, sampling from the sample (!) is an estimate of sampling from the population. Called a bootstrap sample.
• Sample from sample with replacement, or else you get original sample back.

Blue Jays attendances:

ggplot(jays, aes(sample = attendance)) + 
  stat_qq() + stat_qq_line()

• \(t\) procedure for the mean may not be a good idea because the distribution is skewed.
• Previously: hand-waving with sample size.

What actually matters

• It’s not the distribution of the data that has to be approx normal (for a \(t\) procedure).
• What matters is the sampling distribution of the sample mean.
• If the sample size is large enough, the sampling distribution will be normal enough even if the data distribution is not.
  • This is why we had to consider the sample size as well as the shape.
• But how do we know whether this is the case or not? We only have one sample.
• Use the bootstrap to simulate sampling distribution.

Simulating the sampling distribution of sample statistic

• Sample from our sample with replacement.
• Calculate statistic
• Repeat many times (simulation).
• This gives an idea of how our statistic might vary in repeated samples: that is, its sampling distribution.
• Called the bootstrap distribution of the statistic.
• If the bootstrap distribution is approx normal, infer that the true sampling distribution also approx normal, therefore inference about the mean such as \(t\) is good enough.
• If not, we should be more careful.

Bootstrapping the Blue Jays attendances

• Sampling with replacement is done like this (the default sample size is as long as the original data):

boot <- sample(jays$attendance, replace=TRUE)
sort(jays$attendance)

 [1] 14184 14433 15062 15086 15168 15606 16402 17264 17276 18581 19014 19217
[13] 21195 21312 21397 21519 29306 30430 33086 34743 37929 42419 42917 44794
[25] 48414

sort(boot)

 [1] 14184 14433 14433 15086 15086 15168 15168 16402 17264 17264 17276 17276
[13] 17276 17276 18581 19014 21312 21519 33086 37929 42419 42419 42917 42917
[25] 48414

mean(boot)

[1] 23764.76

• That’s one bootstrapped mean. We need a whole bunch.

A whole bunch

• We are now doing a simulation. I like 10,000 samples when testing for normality:

tibble(sim = 1:10000) %>% 
  rowwise() %>% 
  mutate(boot_sample = list(sample(jays$attendance, replace = TRUE))) %>% 
  mutate(my_mean = mean(boot_sample)) -> samples
samples

• for each row:
  • obtain a bootstrap sample (list because we are saving the whole sample in one cell of the dataframe)
  • work out the mean of that bootstrap sample.

Sampling distribution of sample mean

ggplot(samples, aes(x=my_mean)) + geom_histogram(bins=10)

• Is that a slightly long right tail?

Normal quantile plot

might be better than a histogram:

ggplot(samples, aes(sample = my_mean)) + 
  stat_qq()+stat_qq_line()

• a very very slight right-skewness, but very close to normal.
• hence the \(t\)-test is fine for the Blue Jays attendances.

Kids learning to read

• Normal quantile plots, one for each sample:

ggplot(kids, aes(sample = score)) + 
  stat_qq() + stat_qq_line() +
  facet_wrap(~ group)

• These both look close to normal.

Control group

• Pull out control group children
• Obtain bootstrap sampling distribution of scores

kids %>% filter(group == "c") -> controls
tibble(sim = 1:10000) %>% 
  rowwise() %>% 
  mutate(my_sample = list(sample(controls$score, replace = TRUE))) %>% 
  mutate(my_mean = mean(my_sample)) -> samples1
samples1

The bootstrap sampling distribution

ggplot(samples1, aes(sample = my_mean)) +
  stat_qq() + stat_qq_line()

• Close to normal

Same, for treatment group

kids %>% filter(group == "t") -> treated
tibble(sim = 1:10000) %>% 
  rowwise() %>% 
  mutate(my_sample = list(sample(treated$score, replace = TRUE))) %>% 
  mutate(my_mean = mean(my_sample)) -> samples2

The bootstrap sampling distribution

ggplot(samples2, aes(sample = my_mean)) +
  stat_qq() + stat_qq_line()

• Very slightly left-skewed, but close to normal. Not a problem.

Pain relief

• With matched pairs, assumption is of normality of differences:

pain %>% mutate(diff = druga - drugb) -> pain
pain

Bootstrap sampling distribution of differences

tibble(sim = 1:10000) %>% 
  rowwise() %>% 
  mutate(my_sample = list(sample(pain$diff, replace = TRUE))) %>% 
  mutate(my_mean = mean(my_sample)) -> samples
samples

Assess it for normality

ggplot(samples, aes(sample = my_mean)) +
  stat_qq() + stat_qq_line()

• Very skewed to the left (because of low outlier)
• Matched pairs \(t\) not to be trusted at all.

Histogram with many bins

ggplot(samples, aes(x = my_mean)) + geom_histogram(bins = 60)

• actually a very multimodal distribution: one mode for each time the low outlier appears in the bootstrap sampling distribution (can be none at all up to several times, because bootstrap sampling is with replacement).